Note this definition of the Weil pairing is not suitable for
practical computations as the representations
of the functions $g_T(P), g_T(P+S)$
grow quickly with $m$. (There are $2m^2$ poles and zeroes
for each function, which means each function is a product
of $2m^2$ line equations.)
Fortunately an alternative definition
of the Weil pairing lends itself well to explicit computation.

Pullback of Divisors

This is another way to view this definition of the Weil pairing.

Suppose $\alpha$ is an endomorphism, and $g$ is a rational function.
Then a natural construct is to compose $g$ and $\alpha$,
i.e. $g \cdot \alpha$.

For example, if $\alpha$ is translation by a point $T$, then
$g \cdot \alpha (P)= g(P+T)$.

The map $\alpha$ also induces a map on the divisors
$\alpha^* : Div(E) \rightarrow Div(E)$ that takes
the divisor of $g$ to the divisor of $g \cdot \alpha$.